Actividades 6. Resuelve las siguientes ecuaciones por el método de igualación: \( \begin{array}{ll}\left.\begin{array}{r}2 x+y=18 \\ \left.\text { a) } \begin{array}{r}4 x-y=0\end{array}\right\}\end{array} \begin{array}{r}x+12 y=-3 \\ x+y=8\end{array}\right\} & \left.\text { c) } \begin{array}{r}5 x+y=10 \\ 2 x-y=4\end{array}\right\} \\ \left.\begin{array}{ll}2 x+y \\ \left.\text { b) } \begin{array}{r}x-2 y=-2 \\ 2 x+y=7 \\ x+y=4\end{array}\right\} & \left.\text { d) } \begin{array}{r}2 x-y=1\end{array}\right\}\end{array} \begin{array}{r}x+y=3 \\ 2 x-y=0\end{array}\right\}\end{array} \)

Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}2x+y=18\\4x-y=0\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}y=18-2x\\4x-y=0\end{array}\right.\) - step2: Substitute the value of \(y:\) \(4x-\left(18-2x\right)=0\) - step3: Simplify: \(6x-18=0\) - step4: Move the constant to the right side: \(6x=0+18\) - step5: Remove 0: \(6x=18\) - step6: Divide both sides: \(\frac{6x}{6}=\frac{18}{6}\) - step7: Divide the numbers: \(x=3\) - step8: Substitute the value of \(x:\) \(y=18-2\times 3\) - step9: Simplify: \(y=12\) - step10: Calculate: \(\left\{ \begin{array}{l}x=3\\y=12\end{array}\right.\) - step11: Check the solution: \(\left\{ \begin{array}{l}x=3\\y=12\end{array}\right.\) - step12: Rewrite: \(\left(x,y\right) = \left(3,12\right)\) Solve the system of equations \( 5 x+y=10;2 x-y=4 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}5x+y=10\\2x-y=4\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}y=10-5x\\2x-y=4\end{array}\right.\) - step2: Substitute the value of \(y:\) \(2x-\left(10-5x\right)=4\) - step3: Simplify: \(7x-10=4\) - step4: Move the constant to the right side: \(7x=4+10\) - step5: Add the numbers: \(7x=14\) - step6: Divide both sides: \(\frac{7x}{7}=\frac{14}{7}\) - step7: Divide the numbers: \(x=2\) - step8: Substitute the value of \(x:\) \(y=10-5\times 2\) - step9: Simplify: \(y=0\) - step10: Calculate: \(\left\{ \begin{array}{l}x=2\\y=0\end{array}\right.\) - step11: Check the solution: \(\left\{ \begin{array}{l}x=2\\y=0\end{array}\right.\) - step12: Rewrite: \(\left(x,y\right) = \left(2,0\right)\) Solve the system of equations \( 2 x-y=1;x+y=3;2 x-y=0 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}2x-y=1\\x+y=3\\2x-y=0\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}y=-1+2x\\x+y=3\\2x-y=0\end{array}\right.\) - step2: Substitute the value of \(y:\) \(\left\{ \begin{array}{l}x-1+2x=3\\2x-\left(-1+2x\right)=0\end{array}\right.\) - step3: Simplify: \(\left\{ \begin{array}{l}3x-1=3\\1=0\end{array}\right.\) - step4: Calculate: \(\left\{ \begin{array}{l}x \in \varnothing \\y \in \varnothing \end{array}\right.\) - step5: Rewrite: \((x, y) \in \varnothing\) Solve the system of equations \( x-2 y=-2;2 x+y=7;x+y=4 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}x-2y=-2\\2x+y=7\\x+y=4\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}x=-2+2y\\2x+y=7\\x+y=4\end{array}\right.\) - step2: Substitute the value of \(x:\) \(\left\{ \begin{array}{l}2\left(-2+2y\right)+y=7\\-2+2y+y=4\end{array}\right.\) - step3: Simplify: \(\left\{ \begin{array}{l}-4+5y=7\\-2+3y=4\end{array}\right.\) - step4: Solve the equation: \(\left\{ \begin{array}{l}y=\frac{11}{5}\\-2+3y=4\end{array}\right.\) - step5: Substitute the value of \(y:\) \(-2+3\times \frac{11}{5}=4\) - step6: Simplify: \(\frac{23}{5}=4\) - step7: Calculate: \(\left\{ \begin{array}{l}x \in \varnothing \\y \in \varnothing \end{array}\right.\) - step8: Rewrite: \((x, y) \in \varnothing\) Aquí están las soluciones para cada uno de los sistemas de ecuaciones utilizando el método de igualación: ### a) Sistema de ecuaciones: \[ \begin{align*} 1) & \quad 2x + y = 18 \\ 2) & \quad 4x - y = 0 \end{align*} \] **Solución:** Resolviendo el sistema, encontramos: \[ (x, y) = (3, 12) \] ### b) Sistema de ecuaciones: \[ \begin{align*} 1) & \quad x - 2y = -2 \\ 2) & \quad 2x + y = 7 \\ 3) & \quad x + y = 4 \end{align*} \] **Solución:** Este sistema no tiene solución: \[ (x, y) \in \varnothing \quad \text{(No hay solución)} \] ### c) Sistema de ecuaciones: \[ \begin{align*} 1) & \quad 5x + y = 10 \\ 2) & \quad 2x - y = 4 \end{align*} \] **Solución:** Resolviendo el sistema, encontramos: \[ (x, y) = (2, 0) \] ### d) Sistema de ecuaciones: \[ \begin{align*} 1) & \quad 2x - y = 1 \\ 2) & \quad x + y = 3 \\ 3) & \quad 2x - y = 0 \end{align*} \] **Solución:** Este sistema no tiene solución: \[ (x, y) \in \varnothing \quad \text{(No hay solución)} \] ### Resumen de soluciones: - a) \( (3, 12) \) - b) No hay solución - c) \( (2, 0) \) - d) No hay solución